Implicit numerical schemes for generalized heat conduction equations

Rieth, Ágnes; Kovács, Róbert; Fülöp, Tamás

doi:10.48550/arxiv.1805.01108

Cited by 1 publication

(1 citation statement)

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“…The non-dimensional model ( 19)-( 25) is numerically solved using a semi-implicit finite difference method on a staggered grid [57]. The transformations ξ = r/R(t) and η = (r − R(t))/(1 − R(t)) are used to map the evolving domains 0 < r < R(t) and R(t) < r < 1 to stationary domains 0 < η, ξ < 1, respectively.…”

Section: Methodsmentioning

confidence: 99%

Modelling ultra-fast nanoparticle melting with the Maxwell–Cattaneo equation

Hennessy

Calvo-Schwarzwälder

Myers

2019

Applied Mathematical Modelling

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The role of thermal relaxation in nanoparticle melting is studied using a mathematical model based on the Maxwell-Cattaneo equation for heat conduction. The model is formulated in terms of a two-phase Stefan problem. We consider the cases of the temperature profile being continuous or having a jump across the solid-liquid interface. The jump conditions are derived from the sharp-interface limit of a phase-field model that accounts for variations in the thermal properties between the solid and liquid. The Stefan problem is solved using asymptotic and numerical methods. The analysis reveals that the Fourier-based solution can be recovered from the classical limit of zero relaxation time when either boundary condition is used.However, only the jump condition avoids the onset of unphysical "supersonic" melting, where the speed of the melt front exceeds the finite speed of heat propagation. These results conclusively demonstrate that the jump condition, not the continuity condition, is the most suitable for use in models of phase change based on the Maxwell-Cattaneo equation. Numerical investigations show that thermal relaxation can increase the time required to melt a nanoparticle by more than a factor of ten. Thus, thermal relaxation is an important process to include in models of nanoparticle melting and is expected to be relevant in other rapid phase-change processes.can be compared against experimental data and used to assess new theories of nanoscale heat transport and phase change. From a practical perspective, nanoparticles play fundamental roles in novel drug delivery systems [3], materials with modified properties [4,5], and for improving the efficiency of solar collectors [6].Many current and future applications of nanoparticles require quantitative knowledge of how they respond to their thermal environment and their behaviour during melting.The thermal response of a nanoparticle differs from that of a macroscopic body for two main reasons.Firstly, the large ratio of surface to bulk atoms leads to many key thermophysical properties, such as melt temperature [7-9], latent heat [10][11][12], and surface energy [13] becoming dependent on the size of the nanoparticle. Secondly, the mechanism of thermal transport changes between the macroscale and the nanoscale. At the macroscale, heat is transported by a diffusive process that is driven by frequent collisions between thermal energy carriers known as phonons. Diffusive transport of heat across macroscopic length scales is well described by Fourier's law. On nanometer length scales, thermal energy is transported by a ballistic process driven by infrequent collisions between phonons. The finite time between phonon collisions results in a wave-like propagation of heat with finite speed [14,15]. Since Fourier's law leads to an infinite speed of heat propagation, it is not suitable for describing ballistic energy transport.Recent theoretical studies of nanoparticle melting have investigated the role of size-dependent material properties [16][17][18][19][20][21][22] u...

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Section: Methodsmentioning

confidence: 99%